let r be Real; :: thesis: ( - (sqrt 2) <= r & r <= - 1 implies ( sin . (arcsec2 r) = - ((sqrt ((r ^2 ) - 1)) / r) & cos . (arcsec2 r) = 1 / r ) )
(3 / 4) * PI <= PI by Lm6, XXREAL_1:2;
then A1: (3 / 4) * PI in [.0 ,PI .] ;
A2: dom (sec | [.((3 / 4) * PI ),PI .]) c= dom sec by RELAT_1:89;
set x = arcsec2 r;
assume that
A3: - (sqrt 2) <= r and
A4: r <= - 1 ; :: thesis: ( sin . (arcsec2 r) = - ((sqrt ((r ^2 ) - 1)) / r) & cos . (arcsec2 r) = 1 / r )
r in [.(- (sqrt 2)),(- 1).] by A3, A4;
then A5: arcsec2 r in dom (sec | [.((3 / 4) * PI ),PI .]) by Lm30, Th86;
A6: r = (cos ^ ) . (arcsec2 r) by A3, A4, Th90
.= 1 / (cos . (arcsec2 r)) by A5, A2, RFUNCT_1:def 8 ;
PI in [.0 ,PI .] ;
then [.((3 / 4) * PI ),PI .] c= [.0 ,PI .] by A1, XXREAL_2:def 12;
then A7: sin . (arcsec2 r) >= 0 by A5, Lm30, COMPTRIG:24;
- r >= - (- 1) by A4, XREAL_1:26;
then (- r) ^2 >= 1 ^2 by SQUARE_1:77;
then A8: (r ^2 ) - 1 >= 0 by XREAL_1:50;
((sin . (arcsec2 r)) ^2 ) + ((cos . (arcsec2 r)) ^2 ) = 1 by SIN_COS:31;
then (sin . (arcsec2 r)) ^2 = 1 - ((cos . (arcsec2 r)) ^2 )
.= 1 - ((1 / r) * (1 / r)) by A6
.= 1 - (1 / (r ^2 )) by XCMPLX_1:103
.= ((r ^2 ) / (r ^2 )) - (1 / (r ^2 )) by A4, XCMPLX_1:60
.= ((r ^2 ) - 1) / (r ^2 ) ;
then sin . (arcsec2 r) = sqrt (((r ^2 ) - 1) / (r ^2 )) by A7, SQUARE_1:def 4
.= (sqrt ((r ^2 ) - 1)) / (sqrt (r ^2 )) by A4, A8, SQUARE_1:99
.= (sqrt ((r ^2 ) - 1)) / (- r) by A4, SQUARE_1:90
.= - ((sqrt ((r ^2 ) - 1)) / r) by XCMPLX_1:189 ;
hence ( sin . (arcsec2 r) = - ((sqrt ((r ^2 ) - 1)) / r) & cos . (arcsec2 r) = 1 / r ) by A6; :: thesis: verum